Question about topology on $K^\times$ in local CFT I'm trying to parse a page in Milne's CFT notes. The local reciprocity law gives us isomorphisms $$\phi_{L/K}:K^\times/Nm(L^\times)\to \textrm{Gal}(L/K)$$ for all abelian extensions $L$ of a nonarchimedean local field $K$. Taking limits we get by infinite Galois theory the isomorphism $$\phi_K:\widehat{K}^\times\to \textrm{Gal}(K^{ab}/K).$$
If we fix a uniformizer/prime element of $K$, call it $\pi$, then we have the canonical decomposition $$K^\times \simeq \pi^\mathbb{Z}\times U_K,$$ where $U_K$ are the units in the valuation ring of $K$. When we take limits, then we apparently get the decomposition $$\widehat{K}^\times \simeq \pi^\widehat{\mathbb{Z}}\times U_K.$$
My question is really how one should think about $\pi^\widehat{\mathbb{Z}}$. For example $\phi_K(\pi)$ acts as the Frobenius element on $K^{un}$ and it's easy to see how $\phi(\pi^a)$ acts for $a\in\mathbb{Z}$. My questions are the following:


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*How should one think of general exponents in $\widehat{\mathbb{Z}}$ instead of $\mathbb{Z}$?

*How would one show that the fixed field of $\phi_K(\pi)$ i.e. $K_\pi$ is also fixed by $\pi^{\widehat{\mathbb{Z}}}$, so that infinite Galois theory gives us $K^{ab}=K_\pi\cdot K^{un}$?
 A: *

*Here $\pi^\hat{\mathbb{Z}}$ is a notation to denote the group $\hat{\mathbb{Z}}$, but instead of writing an element $n \in \hat{\mathbb{Z}}$ we write $\pi^n$.


In a more general context : let $G$ be a profinite group and $\alpha \in G$. Then we have the following : the map $\mathbb{Z} \rightarrow G, n \mapsto \alpha^n$ extends constinuously to a map $\hat{\mathbb{Z}} \rightarrow G$, and its image is denoted $\pi^\hat{\mathbb{Z}}$ and the image of $n \in \hat{\mathbb{Z}}$ is written $\alpha^n$. Moreover $\pi^\hat{\mathbb{Z}}$ is the closed subgroup generated by $\alpha$.


*

*When $\pi^{\hat{Z}}$ is seen as a subgroup of $Gal(\overline{K}/K)$, then it is the closed subgroup generated by the Frobenius (= image of $\pi$). Since the stabalizer of a subfield is a closed subgroup, this answers your question. 

A: user10676 has basically answered your question.  Maybe I can just add that $K_{\pi}$ is a particular totally ramified abelian extension of $K$ obtained by adjoining the coordinates of all the $p$-power division points in the Lubin--Tate formal group corresponding to $\pi$.
E.g. if $K = \mathbb Q_p$ and $\pi = p$, then the corresponding Lubin--Tate formal group is just the multiplicative formal group, and $K_{\pi}$ is the extension
of $\mathbb Q_p$ obtained by adjoining all $p$-power roots of unity.
